Lawvere's discovery did not appear in print until
Lawvere–Menni [1]} in 2010. In that work the
authors took an important step towards explaining the universal
property by lifting the construction of the Hopf algebra of Möbius
intervals to the objective level. This means that its
comultiplication is realised as something called a pro-comonoidal
structure on certain extensive categories. The original Hopf algebra
is exhibited as being only a numerical shadow of this
categorical construction. There are at least two precursors to the
idea of a more objective approach to incidence algebras. One is
given by Joyal [2]}. In his foundational paper on
species, there is a final section where he considers certain
decomposition structures on categories (that final section has
little to do with species). Another is in the work of
Dür [3]} who constructed incidence coalgebras of
certain categorical and simplicial structures.
